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Mirrors > Home > NFE Home > Th. List > sspsstr | GIF version |
Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.) |
Ref | Expression |
---|---|
sspsstr | ⊢ ((A ⊆ B ∧ B ⊊ C) → A ⊊ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspss 3368 | . 2 ⊢ (A ⊆ B ↔ (A ⊊ B ∨ A = B)) | |
2 | psstr 3373 | . . . . 5 ⊢ ((A ⊊ B ∧ B ⊊ C) → A ⊊ C) | |
3 | 2 | ex 423 | . . . 4 ⊢ (A ⊊ B → (B ⊊ C → A ⊊ C)) |
4 | psseq1 3356 | . . . . 5 ⊢ (A = B → (A ⊊ C ↔ B ⊊ C)) | |
5 | 4 | biimprd 214 | . . . 4 ⊢ (A = B → (B ⊊ C → A ⊊ C)) |
6 | 3, 5 | jaoi 368 | . . 3 ⊢ ((A ⊊ B ∨ A = B) → (B ⊊ C → A ⊊ C)) |
7 | 6 | imp 418 | . 2 ⊢ (((A ⊊ B ∨ A = B) ∧ B ⊊ C) → A ⊊ C) |
8 | 1, 7 | sylanb 458 | 1 ⊢ ((A ⊆ B ∧ B ⊊ C) → A ⊊ C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 357 ∧ wa 358 = wceq 1642 ⊆ wss 3257 ⊊ wpss 3258 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-pss 3261 |
This theorem is referenced by: sspsstrd 3377 |
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